I haven't been to Hampton Court Palace, but I want to say that the National Portrait Gallery is not just a collection of dusty English peers. Like the Portrait Gallery in Washington D.C., it's at a level far above the average English portrait gallery. I have been to the Dulwich Picture Gallery, and their collection of portraits is huge but lame. I happen to have an interest in portraits of actors, but that's a small subset at Dulwich. It takes too long to get to Dulwich and back for your schedule.
I'm not as fond of tapestries, so I can't comment on that. Have you previously been to the V&A? It's not all decorative arts, and it's a very fine collection, with many special exhibitions, most of the time.
There's plenty wrong with Tate Modern, but if you have an interest in modern art, the slam is off-base. I had always wanted to see Mark Rothko's "Seagram House" suite, and he readily gave it to the Tate. Although smaller, this collection is on the level of the Pompidou. If you don't care for modern art, of course skip it. I'm sorry I missed Ai Weiwei's ceramic seeds.